Calculus Examples

$lim_{x \to x} \sqrt{\frac{3 s^{7} - 4 s^{5}}{2 s^{7} + 1}}$

Step 1

Evaluate the limit of $\sqrt{\frac{3 s^{7} - 4 s^{5}}{2 s^{7} + 1}}$ which is constant as $x$ approaches $x$ .

$\sqrt{\frac{3 s^{7} - 4 s^{5}}{2 s^{7} + 1}}$

Step 2

Simplify the answer.

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Step 2.1

Factor $s^{5}$ out of $3 s^{7} - 4 s^{5}$ .

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Step 2.1.1

Factor $s^{5}$ out of $3 s^{7}$ .

$\sqrt{\frac{s^{5} (3 s^{2}) - 4 s^{5}}{2 s^{7} + 1}}$

Step 2.1.2

Factor $s^{5}$ out of $- 4 s^{5}$ .

$\sqrt{\frac{s^{5} (3 s^{2}) + s^{5} \cdot - 4}{2 s^{7} + 1}}$

Step 2.1.3

Factor $s^{5}$ out of $s^{5} (3 s^{2}) + s^{5} \cdot - 4$ .

$\sqrt{\frac{s^{5} (3 s^{2} - 4)}{2 s^{7} + 1}}$

Step 2.2

Rewrite $\sqrt{\frac{s^{5} (3 s^{2} - 4)}{2 s^{7} + 1}}$ as $\frac{\sqrt{s^{5} (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$ .

$\frac{\sqrt{s^{5} (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Rewrite $s^{5} (3 s^{2} - 4)$ as ${(s^{2})}^{2} (s (3 s^{2} - 4))$ .

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Step 2.3.1.1

Factor out $s^{4}$ .

$\frac{\sqrt{s^{4} s (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$

Step 2.3.1.2

Rewrite $s^{4}$ as ${(s^{2})}^{2}$ .

$\frac{\sqrt{{(s^{2})}^{2} s (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$

Step 2.3.1.3

Add parentheses.

$\frac{\sqrt{{(s^{2})}^{2} (s (3 s^{2} - 4))}}{\sqrt{2 s^{7} + 1}}$

Step 2.3.2

Pull terms out from under the radical.

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$

Step 2.4

Multiply $\frac{s^{2} \sqrt{s (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$ by $\frac{\sqrt{2 s^{7} + 1}}{\sqrt{2 s^{7} + 1}}$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}} \cdot \frac{\sqrt{2 s^{7} + 1}}{\sqrt{2 s^{7} + 1}}$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{s^{2} \sqrt{s (3 s^{2} - 4)}}{\sqrt{2 s^{7} + 1}}$ by $\frac{\sqrt{2 s^{7} + 1}}{\sqrt{2 s^{7} + 1}}$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{\sqrt{2 s^{7} + 1} \sqrt{2 s^{7} + 1}}$

Step 2.5.2

Raise $\sqrt{2 s^{7} + 1}$ to the power of $1$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{\sqrt{2 s^{7} + 1}}^{1} \sqrt{2 s^{7} + 1}}$

Step 2.5.3

Raise $\sqrt{2 s^{7} + 1}$ to the power of $1$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{\sqrt{2 s^{7} + 1}}^{1} {\sqrt{2 s^{7} + 1}}^{1}}$

Step 2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{\sqrt{2 s^{7} + 1}}^{1 + 1}}$

Step 2.5.5

Add $1$ and $1$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{\sqrt{2 s^{7} + 1}}^{2}}$

Step 2.5.6

Rewrite ${\sqrt{2 s^{7} + 1}}^{2}$ as $2 s^{7} + 1$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 s^{7} + 1}$ as ${(2 s^{7} + 1)}^{\frac{1}{2}}$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{({(2 s^{7} + 1)}^{\frac{1}{2}})}^{2}}$

Step 2.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{(2 s^{7} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{(2 s^{7} + 1)}^{\frac{2}{2}}}$

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{(2 s^{7} + 1)}^{\frac{2}{2}}}$

Step 2.5.6.4.2

Rewrite the expression.

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{{(2 s^{7} + 1)}^{1}}$

Step 2.5.6.5

Simplify.

$\frac{s^{2} \sqrt{s (3 s^{2} - 4)} \sqrt{2 s^{7} + 1}}{2 s^{7} + 1}$

Step 2.6

Combine using the product rule for radicals.

$\frac{s^{2} \sqrt{(2 s^{7} + 1) s (3 s^{2} - 4)}}{2 s^{7} + 1}$

Step 2.7

Reorder factors in $\frac{s^{2} \sqrt{(2 s^{7} + 1) s (3 s^{2} - 4)}}{2 s^{7} + 1}$ .

$\frac{s^{2} \sqrt{s (2 s^{7} + 1) (3 s^{2} - 4)}}{2 s^{7} + 1}$